1. Field of the Invention
The present invention relates to a method of setting a measuring range in carrying out reciprocal-space mapping of X-ray diffraction measurement.
2. Description of the Related Art
The reciprocal-space mapping measurement is one of the measuring techniques of X-ray diffraction. First of all, a scattering vector will be explained before explanation of the reciprocal-space mapping. FIG. 1 is a view for explaining the scattering vector of X-ray diffraction, in which an X-ray 12 is incident on the surface of a sample 10, and a diffracted X-ray 14 goes out of the sample surface. An angle of the incident X-ray 12 to the surface of the sample 10 is referred to as an incident angle which is denoted by ω, and an angle of the diffracted X-ray 14 to the incident angle 12 is referred to as a diffraction angle which is denoted by 2θ. In the X-ray diffraction measurement, the incident X-ray 12 comes from an X-ray source 16, and the diffracted X-ray 14 is to be detected by an X-ray detector 18.
An X-ray diffraction phenomenon will be explained with the use of the reciprocal space of the crystal which makes up the sample 10. A unit vector S0 is taken as extending in the direction of the incident X-ray 12, and another unit vector S is taken as extending in the direction of the diffracted X-ray 14. Assuming that an incident X-ray vector is defined as S0/λ and a diffracted X-ray vector is defined as S/λ where λ is the wavelength of an X-ray, vectorial subtraction of the incident X-ray vector from the diffracted X-ray vector becomes, as well known, the scattering vector H. The X-ray diffraction principle suggests that when the tip location of the scattering vector H coincides with any lattice point in the reciprocal space, X-ray diffraction occurs at the real lattice plane 20, which is a crystal lattice plane in the real space, corresponding to the reciprocal lattice point. The scattering vector H has the property that the direction is perpendicular to the real lattice plane 20 and the magnitude is equal to the inverse number of the lattice spacing of the real lattice plane 20. The direction of the scattering vector H can be expressed by a tilt angle α to the normal 22 of the surface of the sample 10 within a plane including the incident X-ray 12 and the diffracted X-ray 14.
The reciprocal-space mapping is defined as how the X-ray diffraction intensity varies with ω and 2θ. The mapping can be obtained with the procedure in which the incident angle ω and the diffraction angle 2θ are changed so as to change the direction and the magnitude of the scattering vector within a desired range and the X-ray diffraction measurement is carried out for each scattering vector. How the scattering vector varies with ω and 2θ will be explained below.
FIG. 2 is an explanatory view of the direction change of the scattering vector H with the magnitude unchanged. The incident X-ray vector will be referred to as simply the incident X-ray 12 and the diffracted X-ray vector will be referred to as simply the diffracted X-ray 14 in the description below. It is assumed that the incident angle ω of the incident X-ray 12 is increased by Δω, that is clockwise turning in FIG. 2, and the direction of the diffracted X-ray 14, i.e., the angular location of an X-ray detector, is changed by Δω in the same turning direction. In this case, the direction of the scattering vector H changes while the diffraction angle 2θ does not change. The tilt angle α becomes α+Δω. Thus, if the incident X-ray 12 and the diffracted X-ray 14 are changed in the same turning direction by the same angle as described above, only the direction of the scattering vector H changes. This movement with such angular changes is referred to as an ω scan.
Another scanning method will be explained with reference to FIG. 3 which is an explanatory view of the magnitude change of the scattering vector H with the direction unchanged. When the diffracted X-ray 14 is turned counterclockwise in FIG. 3 by a certain angle, which equals to Δ2θ/2, and the incident X-ray 12 is turned inversely, i.e., clockwise in FIG. 3, by the same angle, the diffraction angle 2θ is changed to 2θ+Δ2θ and the incident angle ω is changed to ω+Δ2θ/2. Thus, if the directions of the incident X-ray and the diffracted X-ray are changed inversely as described above, only the magnitude of the scattering vector H changes with the direction unchanged. This movement with such angular changes is referred to as a 2θ/ω scan.
An operation of the ω scan brings only an ω change with 2θ unchanged, while an operation of the 2θ/ω scan brings a 2θ change along with an ω change which is a half of the 2θ change. The property of the 2θ/ω scan described above deeply concerns the problem between the designation of the relative angle and the designation of the absolute angle in the present invention.
FIG. 4 shows the movement of the scattering vector in the reciprocal space in the ω scan. Each of the tip location of the scattering vector is represented by a black dot, which is referred to hereinafter as a measuring point. Each operation of X-ray diffraction measurement is to be carried out at each black dot. The center position of the measuring range, i.e., the center position of the reciprocal-space mapping, is assumed to be located at a point O. When the ω scan is carried out with the magnitude of the scattering vector kept the same as that at the point O, the measuring point moves from a point A to a point B. The ω varies from a smaller value to a larger value in the scan. The measuring point in the case moves circumferentially centering on the X-ray irradiation point on the sample 10. If it is desired to change the magnitude of the scattering vector to another value for another ω scan, the measuring point is moved, for example, from the point O to a point C, that is the magnitude of the scattering vector is decreased so that 2θ/ω is changed by a certain value. Then the ω scan is carried out with the magnitude of the scattering vector kept the same as that at the point C, i.e., the measuring point moves from a point E to a point F. In the actual procedure, the magnitude of the scattering vector, which corresponds to the value of 2θ/ω, is changed stepwise at certain measuring intervals from the point C to a point D, and the ω scan is carried out for each magnitude of the scattering vector. FIG. 4 shows, for easier understanding, five values of the magnitude of the scattering vector and thus five kinds of the ω scan. An operation of X-ray diffraction measurement is carried out, in each ω scan, at the five measuring points with the direction of the scattering vector different from each other, obtaining twenty-five measured results. It is noted, however, that a larger number of measuring points would be selected generally in the actual reciprocal-space mapping measurement.
FIG. 5 shows the movement of the scattering vector in the reciprocal space in the 2θ/ω scan. Five kinds of the direction, which correspond to the value of ω, of the scattering vector are selected in this case and thus five kinds of the 2θ/ω scan are carried out. The measuring point in the 2θ/ω scan moves on a line passing through the X-ray irradiation point on the sample 10, because the 2θ/ω scan brings the change of the magnitude of the scattering vector with the direction kept constant.
The explanation about the measuring points having been described above is done with the reference to the reciprocal space, and thus the explanation would be clear. Actual measuring conditions, however, must be designated with the use of the incident angle ω and the diffraction angle 2θ. The measuring range of ω in the ω scan may be designated in either relative angle or absolute angle, affecting the shape of the measuring region as shown in FIG. 4 indicating a fair shape of the measuring region and in FIG. 6 indicating a warped shape of the measuring region. The two measuring regions different from each other will be described in detail below.
FIG. 7 is a graph expressing a measuring region for the ω scan shown in FIG. 4 in the coordinate system made of ω-axis and 2θ/ω-axis. The meaning of 2θ/ω, which is used as ordinate, is an angle 20 in the case where 20 and ω are changed in the interlocking fashion in the 2θ/ω scan. The central measuring point O of the reciprocal-space mapping is assumed to be 60 degrees in 2θ/ω and 30 degrees in ω. The angle 2θ/ω is assumed to vary within a range of ±10 degrees and ω is assumed to vary within a range of ±10 degrees too on the basis of the central measuring point O. When ω is scanned within a range of ±10 degrees with the magnitude of the scattering vector kept constant, i.e., 2θ is constant, on the basis of the measuring point O, the measuring point moves from a point A to a point B. The measuring conditions have the five points: 20, 25, 30, 35 and 40 degrees in ω and 60 degrees in 2θ/ω which is kept constant. The angle ω is thus to vary from 20 to 40 degrees.
When it is desired to carry out another ω scan for another value of 2θ/ω, the magnitude of the scattering vector may be changed from the measuring point O with the direction unchanged, i.e., the tilt angle α is unchanged as shown in FIG. 3. For example, the measuring point jumps from the point O into a point C, 50 degrees in 2θ/ω, and the ω scan is carried out with ω varying within a range of ±10 degrees for this value of 2θ/ω. It should be noted, in this case, that when the measuring point moves from the point O to the point C with the angle α kept constant, the angle 2θ is decreased by 10 degrees and further the angle ω is also decreased by 5 degrees in accordance with the decrease of 2θ. The location of the point C is thus 25 degrees in ω. When the angle ω is changed from the point C within a range of ±10 degrees, the angle ω is to vary between 15 degrees, a point E, and 35 degrees, a point F. It will be seen accordingly that the measuring range between 20 and 40 degrees in ω in the case of 60 degrees in 2θ/ω is indeed different from the measuring range between 15 and 35 degrees in ω in the case of 50 degrees in 2θ/ω. A similar situation will occur in the case of every change of 2θ/ω. After all, as shown in FIG. 11, the measuring range becomes within a range of ±10 degrees in ω centering on a line 24 which is constant in a on the basis of the measuring point O.
Although FIG. 11 shows a graph whose abscissa represents an absolute angle of ω, the absolute angle may be replaced with, as shown in FIG. 9, a relative angle Δω measured from the line 24 (see FIG. 11) which is constant in α. It will be seen that the measuring region in FIG. 9 becomes a fair shape. Therefore, if the measuring range of the ω scan is designated by the relative angle Δω, ±10 degrees in the case above, the measuring region with a fair shape such as shown in FIG. 9 and FIG. 4 can be produced.
Incidentally, there exists a certain apparatus in which the measuring range in ω can be designated with the use of selectively the relative angle or the absolute angle on the setting screen for the measuring conditions of the reciprocal-space mapping. If the measuring range in ω is designated in relative angle, it brings a situation such as shown in FIG. 12. The measuring range in ω can be designated with 20 and 40 degrees: the 20-degree value is 10 degrees lower than the center point O which is 30 degrees in ω, and the 40-degree value is 10 degrees higher than the center point O. It would be no problem when 2θ/ω is 60 degrees. However, there is a problem when 2θ/ω is 50 degrees for example, in which the minimum value in ω is 20 degrees which is minus 5 degrees measured from the point C and the maximum value in ω is 40 degrees which is plus 15 degrees measured from the point C. Although the measuring range is selected to range from 20 to 40 degrees in absolute angle, this measuring range is defined to range from minus 5 to plus 15 degrees in relative angle measured from the point C. Such a measuring range may be expressed, in the reciprocal space, with a line of the ω scan passing through the point C in FIG. 6. Comparing the measuring range in FIG. 6 with the measuring range in FIG. 4, the scan range in ω takes a rightward shift. Assuming that the whole measuring region is transferred in the reciprocal space, it becomes a shaded region shown in FIG. 6. The designation of the measuring range in ω with the use of the absolute angle would bring such a warped measuring range disadvantageously. The abscissa of the graph shown in FIG. 12 can be replaced with Δω so as to make a graph shown in FIG. 13, from which it is seen that the measuring range in Δω takes a different shift depending upon the value of 2θ/ω.
If an operator dares to designate the measuring range in ω with the use of the absolute angle with the understanding of the measuring region such as shown in FIG. 6, i.e., the measuring region in Δω as shown in FIG. 13, in the setting operation for the measuring conditions of the reciprocal-space mapping, it would be no problem. It would be considered, however, that the reciprocal-space mapping measurement with the measuring region such as shown in FIG. 6 has no merit. It is presumed accordingly that the operator would think, without careful consideration, the measuring regions are similar to each other between the two types of designation with the use of the relative angle and the absolute angle. Using the measuring region shown in FIG. 6, the data processing after the measurement must be changed as compared with the case with the measuring region shown in FIG. 4. If the measured result is obtained indeed with the measuring region shown in FIG. 6 but the operator misinterprets that the result is obtained with the measuring region shown in FIG. 4, the data processing would have trouble.
There is some merit to suggestions therefore that the designation in absolute angle should be impossible on the setting screen. There are some cases, however, that the designation in absolute angle is more helpful in understanding the region to be measured than in relative angle. Eventually there exists at present a certain apparatus which makes it possible to designate the measuring range in ω with the use of both the relative angle and the absolute angle selectively.
It is noted that, in setting the measuring range in 2θ/ω, the measuring regions become the same as each other between the designation in relative angle and absolute angle. This will be explained below.
FIG. 10 shows the movement of the 2θ/ω scan in the coordinate system with Δω, which is the relative angle, in abscissa and 2θ/ω in ordinate. The angle 2θ/ω is to move always from 50 to 70 degrees whatever Δω is. The measuring regions in this case become the same as each other between the designation in relative angle and in absolute angle.
FIG. 8 shows a graph which is obtained by a conversion in which operation that the abscissa of the graph shown in FIG. 10 is replaced with ω which is the absolute angle. When the 2θ/ω scan is carried out along any line which is kept constant in α, the angle 2θ/ω is to move always from 50 to 70 degrees. It is seen, from both FIG. 10 and FIG. 8, that even if the expression of 2θ/ω in ordinate is changed from the absolute angle to the relative angle which is Δ2θ/ω centering on 60 degrees, the measuring region will remain as it is.
The prior art against the present invention is disclosed in the following publications: B. D. Cullity, Elements of X-ray Diffraction, Second Edition, Japanese Version, Translated by G. Matsumura, Eighth Reprinted Edition, Issued by Agune (1990) page 445–458, referred to hereinafter as the first publication; Japanese Patent Publication No. 2000-39409 A (2000), referred to hereinafter as the second publication; and Japanese Patent Publication No. 11-304729 A (1999), referred to hereinafter as the third publication. The explanation of the X-ray diffraction phenomenon using the scattering vector is disclosed in the first publication. The reciprocal-space mapping measurement is disclosed in the second and the third publications, which do not mention the difference between the relative angle and the absolute angle in setting the measuring range of the reciprocal-space mapping.